Studying the various notion of curvature, I have not been able to get the intuition and deeper understanding beyond their definitions.

Let me first give the definitions I know. Throughout, I will consider a $m-$dimensional Riemannian manifold $(M,g)$ equipped with Levi-Civita connection $\nabla$.

We have defined *Riemannian curvature tensor* to be the collection of trilinear maps $$R_p:T_pM \times T_pM \times T_pM \to T_pM, \ (u,v,w)\mapsto R(X,Y)Z(p), \quad p\in M$$
where $X,Y,Z$ are vector fields defined on some neighbourhood of $p$ with $X(p)=u,Y(p)=v,Z(p)=w$, and $R(X,Y)Z:=\nabla_Y\nabla_XZ-\nabla_X\nabla_YZ+\nabla_{[X,Y]}Z.$ **This seems to be a purely algebraic defintion; I don't see any geometry here.**

Second is the *sectional curvature.* For $p \in M$, let us take a two-dimensional subspace $E \leq T_pM.$ Suppose $(u,v)$ be the basis of $E$. We then define the *sectional curvature $K(E)$ of $M$ at $p$ with respect to $E$* as $$K(E)=K(u,v):=\frac{\langle R(u,v)v,u\rangle}{\langle u,u\rangle\langle v,v\rangle-\langle u,v\rangle^2}.$$
Third is the *Ricci curvature.* Let $p\in M$ and $x\in T_pM$ be a unit vector. Let $(z_1,\cdots,z_m)$ be an orthonormal basis of $T_pM$ s.t. $z_m=x.$ The *Ricci curvature of $M$ at $p$ with respect of $x$* is $$Ric_p(x):=\frac{1}{m-1}\sum_{i=1}^{m-1}K(x,z_i)$$ where $K(x,z_i)$ is the sectional curvature defined above.

Finally the scalar curvature of $M$ at $p$ is defined to be $$K(p):=\frac{1}{m}\sum_{i=1}^m Ric_p(z_i).$$

My questions are about understanding these four notions beyond the defintions. How should I think of each of them? Are they related to one another in a sense that is one notion of curvature stronger than another? I am sorry if these questions are too much for one post.